We are conducting COLD GASS, a legacy survey for molecular gas in nearby galaxies. Using the IRAM 30-m telescope, we measure the CO(1-0) line in a sample of ~350 nearby (Mpc), massive galaxies (log(M */Msun) > 10.0). The sample is selected purely according to stellar mass, and therefore provides an unbiased view of molecular gas in these systems. By combining the IRAM data with Sloan Digital Sky Survey (SDSS) photometry and spectroscopy, GALEX imaging and high-quality Arecibo Hi data, we investigate the partition of condensed baryons between stars, atomic gas and molecular gas in 0.1-10L* galaxies. In this paper, we present CO luminosities and molecular hydrogen masses for the first 222 galaxies. The overall CO detection rate is 54 per cent, but our survey also uncovers the existence of sharp thresholds in galaxy structural parameters such as stellar mass surface density and concentration index, below which all galaxies have a measurable cold gas component but above which the detection rate of the CO line drops suddenly. The mean molecular gas fractionof the CO detections is 0.066 +/- 0.039, and this fraction does not depend on stellar mass, but is a strong function of (NUV -r) colour. Through stacking, we set a firm upper limit offor red galaxies with NUV -r> 5.0. The average molecular-to-atomic hydrogen ratio in present-day galaxies is 0.3, with significant scatter from one galaxy to the next. The existence of strong detection thresholds in both the Hi and CO lines suggests that 'quenching' processes have occurred in these systems. Intriguingly, atomic gas strongly dominates in the minority of galaxies with significant cold gas that lie above these thresholds. This suggests that some re-accretion of gas may still be possible following the quenching event.